Thanks a lot feynman's trick was a good read. But if suppose i did have to use the given result then, what would be the method to go forward with?Orodruin said:You are supposed to use the knowledge of the result of the given integral to find out the integral you want. Changing the integral to one with ##-x## in the exponent instead of ##-x^2/2## is a good start. I also suggest looking up Feynman's trick of differentiating an integral with respect to a parameter (not the integration variable) and changing the order of integration and differentiation.
I was just trying it i understood how it works. I should be able to use both together. Thanks a lot.Orodruin said:I would use Feynman's trick together with the given result. The alternative is doing partial integration. It all is going to boil down to evaluation of the given integral in the end.
Substitution is used to simplify the integrand and make it easier to solve. In this case, substituting u = √x will eliminate the fractional exponent and make the integral easier to evaluate.
The best substitution to use is one that will eliminate the complex or difficult components of the integrand. In this case, u = √x is a good choice because it eliminates the fractional exponent.
The first step is to choose the appropriate substitution, in this case u = √x. Then, substitute u into the integrand and solve for the new integral in terms of u. After integrating, substitute back in the original variable x to get the final solution.
The constant √2π is a result of the substitution and is known as the Jacobian. It is a factor that is used to account for the change in variables and is often present in integrals solved using substitution.
You can check your solution by taking the derivative of the result and seeing if it matches the original integrand. If it does, then your solution is correct. You can also use a graphing calculator to graph the original integrand and your solution to visually confirm that they are the same.